It is obvious that \(K\) -means and Gaussian Mixture Models (GMMs) are invariant with respect to the shift of data features. Since these methods perform directly on the data features, shifting refers to adding constant \(\alpha \)  to all the features. Under this shift, the centroids (in \(K\) -means) and the means (in GMM) are shifted by \(\alpha \)  as well, but their proportional distances stay the same.
The other parameters, i.e., the clustering assignments (in \(K\) -means), and the assignment probabilities, covariance matrices and weights (in GMM) do not change. One might assume that by shift only the location of the clusters is affected without modifying the cluster memberships.
A similar argument applies to a density-based clustering method such as DBSCAN [1]} wherein shifting data features does not modify the clustering solution, except a consistent shift of the geographical locations of the clusters together.
